Measure Algebras on Homogeneous Spaces
Hossein Javanshiri, Narguess Tavallaei

TL;DR
This paper explores the structure of measure algebras on homogeneous spaces, establishing conditions under which they form Banach algebras with or without involution, and analyzing their amenability.
Contribution
It introduces a convolution on measure spaces of homogeneous spaces, characterizes when these form $*$-Banach algebras, and links amenability to subgroup normality and group amenability.
Findings
$M(G/H)$ is a quotient of $M(G)$ and forms a Banach algebra.
$M(G/H)$ is a $*$-Banach algebra only if $H$ is normal.
$L^1(G/H)$ is a Banach subalgebra of $L^1(G)$ with a right approximate identity.
Abstract
For a locally compact group and a compact subgroup , we show that the Banach space may be considered as a quotient space of . Also, we define a convolution on which makes it into a Banach algebra. It may be identified with a closed subalgebra of the involutive Banach algebra , and there is no involution on compatible with this identification unless is a normal subgroup of . In other words, is a -Banach subalgebra of only if is a normal subgroup of . As well, it is a unital Banach algebra just when is a normal subgroup. Furthermore, when is attached to a strongly quasi-invariant measure, is a Banach subspace of . Using the restriction of the convolution on , we obtain a Banach algebra , which may be considered as a Banach subalgebra of , with a right…
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Operator Algebra Research · Advanced Banach Space Theory
